Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system

Colli, Pierluigi; Frigeri, Sergio; Grasselli, Maurizio

doi:10.1016/j.jmaa.2011.08.008

Cited by 91 publications

(170 citation statements)

References 38 publications

(49 reference statements)

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“…We recall that the latter regularity is readily available for the Navier-Stokes equation (1.6) by the energy estimate performed earlier in Proposition 3.1 (cf. e.g., [8]) whereas in the case of the Euler equation (1.2) much less is true, see (3.1). On the other hand, our argument here makes also use of the vorticity equation (3.7), which exploited in unison with (3.8) can produce the required bounds in (3.13).…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

“…Our assumptions on F, J remain essentially the same as in [8,9,12,11,17], and actually we can require much less than there.…”

Section: We Endowmentioning

confidence: 99%

“…This system assumes the case of matched densities for the two fluids and constant mobility. On the other hand, the system comprising of (1.6), (1.1), (1.3), subject to homogeneous Neumann and no slip boundary conditions for µ and u, respectively, has been analyzed recently in [8,9,10,12,13,11] under various assumptions on F, J and on the mobility and viscosity coefficients, respectively. We also recall that the nonlocal Cahn-Hilliard-Navier-Stokes system described earlier is a generalized version of the classical Cahn-Hilliard-Navier-Stokes system when in the place of aϕ − J * ϕ one usually finds −∆ϕ, see [1,2,5,7,14,15,16,26,27,28] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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On an Inviscid Model for Incompressible Two-Phase Flows with Nonlocal Interaction

Gal

2016

J. Math. Fluid Mech.

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Abstract. We consider a diffuse interface model which describes the motion of an ideal incompressible mixture of two immiscible fluids with nonlocal interaction in two-dimensional bounded domains. This model consists of the Euler equation coupled with a convective nonlocal Cahn-Hilliard equation. We establish the existence of globally defined weak solutions as well as well-posedness results for strong/classical solutions.

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Section: Proofs Of the Main Resultsmentioning

confidence: 99%

“…Our assumptions on F, J remain essentially the same as in [8,9,12,11,17], and actually we can require much less than there.…”

Section: We Endowmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On an Inviscid Model for Incompressible Two-Phase Flows with Nonlocal Interaction

Gal

2016

J. Math. Fluid Mech.

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“…This theory has been widely studied from the physical, the mathematical and the computational points of view Jacqmin, 1999;Gal and Grasselli, 2010;Kay et al, 2008;Boyer et al, 2010;Gal and Grasselli, 2011;Lowengrub and Truskinovsky, 1998;Colli et al, 2012;Kim et al, 2004;Liu and Shen, 2003). To derive the theory, let us assume that we have two immiscible components with volume fractions ϕ 1 and ϕ 2 , respectively.…”

Section: Navier-stokes-cahn-hilliardmentioning

confidence: 99%

Computational Phase‐Field Modeling

Gómez

Zee

2017

Encyclopedia of Computational Mechanics Second Edition

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Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community.

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“…We end this section by recalling a simple result, which will be used in the proofs below (see also [23,Lemma 1]). …”

Section: Notation and Functional Setupmentioning

confidence: 99%

Analysis of a Cahn–Hilliard–Ladyzhenskaya system with singular potential

Bosia

2013

Journal of Mathematical Analysis and Applications

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The evolution of a mixture of two incompressible and (partially) immiscible fluids is here described by Ladyzhenskaya–Navier–Stokes type equations for the (average) fluid velocity coupled with a convective Cahn–Hilliard equation with a singular (e.g., logarithmic) potential. The former is endowed with no-slip boundary conditions, while the latter is subject to no-flux boundary conditions so that the total mass is conserved. Here we first prove the existence of a weak solution in three-dimensions and some regularity properties. Then we establish the existence of a weak trajectory attractor for a sufficiently general time-dependent external force. Finally, taking advantage of the validity of the energy identity, we show that the trajectory attractor actually attracts with respect to the strong topology

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Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system

Cited by 91 publications

References 38 publications

On an Inviscid Model for Incompressible Two-Phase Flows with Nonlocal Interaction

On an Inviscid Model for Incompressible Two-Phase Flows with Nonlocal Interaction

Computational Phase‐Field Modeling

Analysis of a Cahn–Hilliard–Ladyzhenskaya system with singular potential

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